Module 4: Factorial Treatment Structure

Example: No Interaction, interpret Main Effects

Example 4.2: Bakery

Treatment Structure: 3 x 2 Full Factorial

  • Shelf Height (Bottom, Middle, Top)
  • Shelf Width (Regular, Wide)

Design Structure: CRD with r = 2 stores per treatment combination.

Response: Bread Sales

The Treatment Effects Model (Two-way Factorial)

\[y_{ijk}=\mu+\alpha_i+\beta_j+\alpha\beta_{ij}+\epsilon_{ijk} \text{ with } \epsilon_{ijk} \sim \text{ iid }N(0,\sigma^2)\]

\[\text{for } i=1,2,3,…,a; j=1,2,…,b; k=1,2,….,r\]

  • \(y_{ijk}\): is the response (sales) from the \(k^{th}\) experimental unit (store) using the \(i^{th}\) level of Factor A (height) and \(j^{th}\) level of Factor B (width) combination
  • \(\mu\): is the grand/overall mean sales
  • \(\alpha_i\): is the effect of the \(i^{th}\) level of factor A (height)
  • \(\beta_j\): is the effect of the \(j^{th}\) level of factor B (width)
  • \(\alpha\beta_{ij}\): is the interaction effect between the \(i^{th}\) level of A (height) and \(j^{th}\) level of B (width)
  • \(\epsilon_{ijk}\): the experimental error associated with the \(k^{th}\) experimental unit (store) using the \(i^{th}\) level of Factor A (height) and \(j^{th}\) level of Factor B (width) combination

Decision Flowchart

What are we testing?

Interaction

\[H_0:\text{ All } \alpha\beta_{ij} = 0 \text{ vs } H_A: \text{At least one } \alpha\beta_{ij} \ne 0\]

Main Effect of A

\[H_0:\text{ All } \alpha_{i} = 0 \text{ vs } H_A: \text{At least one } \alpha_{i} \ne 0\]

Main Effect of B

\[H_0:\text{ All } \beta_{j} = 0 \text{ vs } H_A: \text{At least one } \beta_{j} \ne 0\]

ANOVA

options(contrasts = c("contr.sum", "contr.poly"))
bakery_mod <- lm(sales ~ height + width + height:width, data = bakery_data)
anova(bakery_mod)
Analysis of Variance Table

Response: sales
             Df Sum Sq Mean Sq F value    Pr(>F)    
height        2   1544  772.00 74.7097 5.754e-05 ***
width         1     12   12.00  1.1613    0.3226    
height:width  2     24   12.00  1.1613    0.3747    
Residuals     6     62   10.33                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Do these results make sense?

Inspecting the interaction plot between height and width on bread sales, does there visually appear to be a significant interaction?

Where should we proceed?

  1. Interaction: We do not have enough evidence of a significant interaction effect between shelf height and shelf width on bread sales (F = 1.16; df = 2,6; p = 0.375).
  2. Width Main: We do not have enough evidence of a significant main effect of shelf width on bread sales (F = 1.16; df = 1,6; p = 0.323).
  3. Height Main: We do have evidence of a significant main effect of height on bread sales (F = 74.7; df = 2, 6; p < 0.0001).

What are the marginal mean sales for each height (sig main effect)? Which heights differ?

R: Marginal Height LSMEANS

library(emmeans)
height_lsmeans <- emmeans(bakery_mod, specs = ~ height) 
NOTE: Results may be misleading due to involvement in interactions
height_lsmeans
 height emmean   SE df lower.CL upper.CL
 bottom     44 1.61  6     40.1     47.9
 middle     67 1.61  6     63.1     70.9
 top        42 1.61  6     38.1     45.9

Results are averaged over the levels of: width 
Confidence level used: 0.95 
emmip(bakery_mod, ~ height, CIs = TRUE, adjust = "tukey") +
  labs(y = "Bread Sales",
       x = "height")

R: Height pairwise comparisons

pairs(height_lsmeans, adjust = "tukey", infer = c(T,T))
 contrast        estimate   SE df lower.CL upper.CL t.ratio p.value
 bottom - middle      -23 2.27  6   -29.97   -16.03 -10.119  0.0001
 bottom - top           2 2.27  6    -4.97     8.97   0.880  0.6714
 middle - top          25 2.27  6    18.03    31.97  10.999 <0.0001

Results are averaged over the levels of: width 
Confidence level used: 0.95 
Conf-level adjustment: tukey method for comparing a family of 3 estimates 
P value adjustment: tukey method for comparing a family of 3 estimates 
library(multcomp)
cld(height_lsmeans, decreasing = F, Letters = LETTERS, adjust = "tukey")
 height emmean   SE df lower.CL upper.CL .group
 top        42 1.61  6     36.7     47.3  A    
 bottom     44 1.61  6     38.7     49.3  A    
 middle     67 1.61  6     61.7     72.3   B   

Results are averaged over the levels of: width 
Confidence level used: 0.95 
Conf-level adjustment: sidak method for 3 estimates 
P value adjustment: tukey method for comparing a family of 3 estimates 
significance level used: alpha = 0.05 
NOTE: If two or more means share the same grouping symbol,
      then we cannot show them to be different.
      But we also did not show them to be the same.

JMP: Main Effect of Height

What can we conclude?

  • For bread placed on the middle shelf, the estimated mean bread sales is 67 (s.e. = 1.61).
  • This is estimated to be 25 more sales than for bread placed on the top shelf (t = 10.99; df = 6; p < 0.0001) and 23 more sales than for bread placed on the bottom shelf (t = 10.12; df = 6; p = 0.0001).